## An update on spaces of embeddings

- Date: 03/05/2008

Ryan Budney (University of Victoria)

University of British Columbia

The Cerf-Morlet comparison theorem states that the group of

diffeomorphisms of the compact n-ball which is the identity on the

boundary is an (n+1)-fold loop space, whose (n+1)-fold de-looping is

the homotopy-quotient of the group of the PL-homeomorphisms of R^n by

the smooth homeomorphisms of R^n. In 2002 I observed that this

Cerf-Morlet iterated loop space structure generalizes to an iterated

loop space structure on the space of embeddings of R^j in R^n which are

standard outside of a fixed ball and equipped with a trivialization of

a tubular neighbourhood. This talk will be about the homotopy-type of

these embedding spaces, viewed as iterated loop-spaces. I've been able

to make the most progress in the case (j,n)=(1,3), which I will

describe in detail. My most recent work on this has to do with a

`realization problem' associated to the (j,n)=(1,3) case which boils

down to a study of the symmetry groups of a certain family of

hyperbolic links in S3 together with a family of natural

representations of these symmetry groups.

Algebra/Topology Seminar 2008